Assume \(\widehat{\theta }_0\)  is the LASSO estimate obtained under proper penalization. Denote \(s<n\)  as the number of nonzero coefficients in \(\theta _0\) . When \({\bf x}\)  follows Gaussian and the noise \(\epsilon \)  satisfies \(\mathbb {E}\exp \lbrace  t\epsilon ^2 \rbrace <\infty \)  for some \(t>0\) , based on [1]}, [2]}, we have with probability tending to 1,
\(\Vert \widehat{\theta }_0-\theta _0\Vert =O\left(\sigma \sqrt{\frac{s\log p}{n}}\right).\) 
